Rainbow planar Tur{á}n numbers of cycles

The rainbow Tur{á}n number of a fixed graph $H$, denoted by ${\text{ex}}^{\ast }(n,H)$, is the maximum number of edges in an $n$-vertex graph such that it admits a proper edge coloring with no rainbow $H$. We study this problem in planar setting. The rainbow planar Tur{á}n number of a graph $H$, denoted by ${{\text{ex}}_{\mathcal{P}}}^{\ast }(n,H)$, is the maximum number of edges in an $n$-vertex planar graph such that it has a proper edge coloring with no rainbow $H$. We consider the rainbow planar Tur{á}n number of cycles. Since $C_3$ is complete, ${{\text{ex}}_{\mathcal{P}}}^{\ast }(n,C_3)$ is exactly its planar Tur{á}n number, which is $2n-4$ for $n\ge 3$. We show that ${{\text{ex}}_{\mathcal{P}}}^{\ast }(n,C_4)=3n-6$ for $n=k^2-3k+2$ where $k\ge 5$, and ${{\text{ex}}_{\mathcal{P}}}^{\ast }(n,C_k)=3n-6$ for all $k\ge 5$ and $n\ge 3$.